My question is in regards to the proof of Lemma 2 in the paper "The Fundamental Theorem of Algebra" for the Quaternions, by Eilenberg and Niven.
http://www.ams.org/journals/bull/1944-50-04/S0002-9904-1944-08125-1/S0002-9904-1944-08125-1.pdf
In the proof, they show that the Jacobian is non-zero for a certain map, however we are working over a non-commutative ring. Usual determinants don't make sense over non-commutative rings, so how is it that we're able to make sense of calculating the determinant?